On Friday, April 29, 2022 at 4:35:05 AM UTC+1, Tim Chow wrote:
XGID=-CB-CBC-aa--b----cbcBba---:1:1:1:42:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| O O | | O X O O |
| O O | | O X O |
| O | | O |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X X | +---+
| O | | X X X X X | | 2 |
| O O O | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 87 O: 126 X-O: 0-0
Cube: 2, X own cube
X to play 42
In a typical holding game, the holder has an anchor and is significantly behind in the pip count. The idea is to hold on to the anchor, build
the board, and wait to hit a shot. However, there are positions which superficially look like a typical holding game, but where the player
with the anchor is ahead in the pip count. Then the roles of holder
and holdee are switched. The player with the anchor is looking for a
good time to make a break for it. Breaking too soon risks getting
attacked. Waiting too long to break may mean wasting racing pips and crunching the board.
Here, X is way ahead in the pip count with the stronger board, and hence
is looking for a chance to make a break for it. Playing something like
6/2 4/2 is bad; O's board is getting stronger and X is not likely to get
a better chance to run. The question is, how should X play the 2?
Running 20/14 looks bad because it looks like everything hits. But it
is XG's preferred play. I think this one is difficult to calculate,
because if you move the blot in O's board by one pip, then 20/16 6/4
becomes XG's favorite (see variant).
1. Rollout¹ 20/14 eq:+0.018
Player: 42.34% (G:18.88% B:0.61%)
Opponent: 57.66% (G:10.75% B:0.36%)
Confidence: ±0.008 (+0.010..+0.026) - [100.0%]
2. Rollout¹ 20/16 6/4 eq:-0.033 (-0.051)
Player: 41.67% (G:16.90% B:0.45%)
Opponent: 58.33% (G:11.87% B:0.42%)
Confidence: ±0.008 (-0.041..-0.025) - [0.0%]
3. Rollout¹ 20/16 4/2 eq:-0.043 (-0.061)
Player: 40.88% (G:17.50% B:0.77%)
Opponent: 59.12% (G:12.39% B:0.41%)
Confidence: ±0.008 (-0.051..-0.036) - [0.0%]
4. Rollout¹ 6/2 4/2 eq:-0.128 (-0.146)
Player: 40.89% (G:10.11% B:0.09%)
Opponent: 59.11% (G:13.05% B:0.57%)
Confidence: ±0.007 (-0.135..-0.120) - [0.0%]
5. Rollout¹ 5/3 5/1 eq:-0.143 (-0.161)
Player: 40.41% (G:9.69% B:0.08%)
Opponent: 59.59% (G:12.81% B:0.47%)
Confidence: ±0.007 (-0.150..-0.136) - [0.0%]
¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
eXtreme Gammon Version: 2.19.207.pre-release
-------
Variant
-------
XGID=-CB-CBC-aa--b----cbcBb-a--:1:1:1:42:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| O O | | O X O O |
| O O | | O X O |
| O | | O |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X X | +---+
| O | | X X X X X | | 2 |
| O O O | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 87 O: 125 X-O: 0-0
Cube: 2, X own cube
X to play 42
1. Rollout¹ 20/16 6/4 eq:+0.167
Player: 48.81% (G:19.36% B:0.57%)
Opponent: 51.19% (G:10.15% B:0.35%)
Confidence: ±0.008 (+0.159..+0.175) - [100.0%]
2. Rollout¹ 20/16 4/2 eq:+0.145 (-0.022)
Player: 47.32% (G:20.23% B:0.86%)
Opponent: 52.68% (G:10.57% B:0.35%)
Confidence: ±0.008 (+0.137..+0.153) - [0.0%]
3. Rollout¹ 20/14 eq:+0.115 (-0.052)
Player: 44.72% (G:23.18% B:1.39%)
Opponent: 55.28% (G:11.16% B:0.34%)
Confidence: ±0.009 (+0.106..+0.124) - [0.0%]
With 20/14, most rolls will hit, yes.
But the rolls we're scared of aren't the hitting rolls,
but the hit-and-cover rolls.
I'll count them (in the original position, not the variant): 11/12/22/23/24/25/33/35/36/45/46/55/56
There are 22 of these, but some of the hit-and-covers allow return
hits on the opponent's 7 point.
The next sequence is more likely than not to benefit the opponent. But, often it doesn't,
and when it doesn't, the opponent is dancing against a 5 point board.
Whereas, where it benefits the opponent, the benefits are smaller.
So we have more-likely-benefit for the opponent vs greater-potential-benefit for us.
Since the equity is around zero, these two factors roughly cancel out.
Clearly moving the opponent's blot from 3 to 2 makes us more wary of being hit because there are fewer return hits on the 2 point than the 3 point -- 12 vs 14.
Paul
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